Question

A population of bears increased by 50% in 4 years. If the situation is modeled by an annual growth rate compounded continuously, which formula could be used to find the annual rate according to the exponential growth function?

Answer 1

Answer:

$A=P(1+\frac{r}{100} )^n$

Therefore the annual rate to the exponential growth function is 10.66%

Step-by-step explanation:

Given that, a population of bears increased by 50% in 4 year.

The growth rate is compound continuously.

So we use the compound formula:

$A=P(1+\frac{r}{100} )^n$

Let the initial population of bear was x.

Since 50% of growth of bear is increased in 4 year.

Therefore the number of  bears increased  $=(x\times \frac{50}{100} )$= 0.5 x in 4 year.

Therefore the total number of bear after 4 years is =(x+0.5x) = 1.5 x

Here A = 1.5x,

P=x,

n=4

Therefore,

$1.5x=x(1+\frac{r}{100})^4$

$\Rightarrow 1.5=(1+\frac{r}{100})^4$     [ cancel x from both sides]

$\Rightarrow \sqrt[4]{ 1.5}=(1+\frac{r}{100})$

$\Rightarrow (1+\frac{r}{100})= 1.1066$

$\Rightarrow \frac{r}{100}= 1.1066-1$

$\Rightarrow \frac{r}{100}= 0.1066$

$\Rightarrow r=10.66$

Therefore the annual rate to the exponential growth function is 10.66%.